Engineering Calculus Notes 215

# Engineering Calculus Notes 215 - i 2 π π dt = i 2 π dt =...

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2.5. INTEGRATION ALONG CURVES 203 As another example, let us use this formalism to compute the circumference of a circle. The circle is not an arc, but the domain of the standard parametrization −→ p ( t ) = (cos t, sin t ) 0 t 2 π can be partitioned via P = { 0 ,π, 2 π } into two semicircles, C i , i = 1 , 2, which meet only at the endpoints; it is natural then to say s ( C ) = s ( C 1 ) + s ( C 2 ) . We can calculate that ˙ x ( t ) = sin t ˙ y ( t ) = cos t so d s = r ( sin t ) 2 + (cos t ) 2 dt = dt and thus s ( C ) = s ( C 1 ) + s ( C 2 ) = i π 0 dt +
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Unformatted text preview: i 2 π π dt = i 2 π dt = 2 π. The example of the circle illustrates the way that we can go from the deFnition of arclength for an arc to arclength for a general curve . By Exercise 11 in § 2.4 , any parametrized curve C can be partitioned into arcs C k , and the arclength of C is in a natural way the sum of the arclengths of these arcs: s ( C ) = s k s ( C k ) ;...
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